\(\int (a+b \arccos (1+d x^2))^{5/2} \, dx\) [64]

Optimal result

Integrand size = 16, antiderivative size = 249 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=-\frac {5 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2}}{d x}+x \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}-\frac {30 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{\left (\frac {1}{b}\right )^{5/2} d x}+\frac {30 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{\left (\frac {1}{b}\right )^{5/2} d x}+\frac {30 b^2 \sqrt {a+b \arccos \left (1+d x^2\right )} \sin ^2\left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{d x} \] Output:

-5*b*(-d^2*x^4-2*d*x^2)^(1/2)*(a+b*arccos(d*x^2+1))^(3/2)/d/x+x*(a+b*arcco 
s(d*x^2+1))^(5/2)-30*Pi^(1/2)*cos(1/2*a/b)*FresnelS((1/b)^(1/2)*(a+b*arcco 
s(d*x^2+1))^(1/2)/Pi^(1/2))*sin(1/2*arccos(d*x^2+1))/(1/b)^(5/2)/d/x+30*Pi 
^(1/2)*FresnelC((1/b)^(1/2)*(a+b*arccos(d*x^2+1))^(1/2)/Pi^(1/2))*sin(1/2* 
a/b)*sin(1/2*arccos(d*x^2+1))/(1/b)^(5/2)/d/x+30*b^2*(a+b*arccos(d*x^2+1)) 
^(1/2)*sin(1/2*arccos(d*x^2+1))^2/d/x
 

Mathematica [A] (verified)

Time = 2.01 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=-\frac {2 \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right ) \left (15 b^{5/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right )-15 b^{5/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )+\sqrt {a+b \arccos \left (1+d x^2\right )} \left (5 a b \cos \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )+\left (a^2-15 b^2\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )+b^2 \arccos \left (1+d x^2\right )^2 \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )+b \arccos \left (1+d x^2\right ) \left (5 b \cos \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )+2 a \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )\right )\right )\right )}{d x} \] Input:

Integrate[(a + b*ArcCos[1 + d*x^2])^(5/2),x]
 

Output:

(-2*Sin[ArcCos[1 + d*x^2]/2]*(15*b^(5/2)*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[Sq 
rt[a + b*ArcCos[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])] - 15*b^(5/2)*Sqrt[Pi]*Fres 
nelC[Sqrt[a + b*ArcCos[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*Sin[a/(2*b)] + Sqrt 
[a + b*ArcCos[1 + d*x^2]]*(5*a*b*Cos[ArcCos[1 + d*x^2]/2] + (a^2 - 15*b^2) 
*Sin[ArcCos[1 + d*x^2]/2] + b^2*ArcCos[1 + d*x^2]^2*Sin[ArcCos[1 + d*x^2]/ 
2] + b*ArcCos[1 + d*x^2]*(5*b*Cos[ArcCos[1 + d*x^2]/2] + 2*a*Sin[ArcCos[1 
+ d*x^2]/2]))))/(d*x)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5314, 5311}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a+b \arccos \left (d x^2+1\right )\right )^{5/2} \, dx\)

\(\Big \downarrow \) 5314

\(\displaystyle -15 b^2 \int \sqrt {a+b \arccos \left (d x^2+1\right )}dx-\frac {5 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^{3/2}}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^{5/2}\)

\(\Big \downarrow \) 5311

\(\displaystyle -15 b^2 \left (-\frac {2 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}+\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sin ^2\left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \sqrt {a+b \arccos \left (d x^2+1\right )}}{d x}\right )-\frac {5 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^{3/2}}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^{5/2}\)

Input:

Int[(a + b*ArcCos[1 + d*x^2])^(5/2),x]
 

Output:

(-5*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2])^(3/2))/(d*x) + x* 
(a + b*ArcCos[1 + d*x^2])^(5/2) - 15*b^2*((2*Sqrt[Pi]*Cos[a/(2*b)]*Fresnel 
S[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[ArcCos[1 + d* 
x^2]/2])/(Sqrt[b^(-1)]*d*x) - (2*Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + 
b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2])/(Sq 
rt[b^(-1)]*d*x) - (2*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[ArcCos[1 + d*x^2]/2 
]^2)/(d*x))
 

Defintions of rubi rules used

rule 5311

Int[Sqrt[(a_.) + ArcCos[1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[-2*Sqrt 
[a + b*ArcCos[1 + d*x^2]]*(Sin[ArcCos[1 + d*x^2]/2]^2/(d*x)), x] + (-Simp[2 
*Sqrt[Pi]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2]*(FresnelC[Sqrt[1/(Pi*b)]*Sq 
rt[a + b*ArcCos[1 + d*x^2]]]/(Sqrt[1/b]*d*x)), x] + Simp[2*Sqrt[Pi]*Cos[a/( 
2*b)]*Sin[ArcCos[1 + d*x^2]/2]*(FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[1 
 + d*x^2]]]/(Sqrt[1/b]*d*x)), x]) /; FreeQ[{a, b, d}, x]
 

rule 5314

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*( 
a + b*ArcCos[c + d*x^2])^n, x] + (-Simp[2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(( 
a + b*ArcCos[c + d*x^2])^(n - 1)/(d*x)), x] - Simp[4*b^2*n*(n - 1)   Int[(a 
 + b*ArcCos[c + d*x^2])^(n - 2), x], x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c 
^2, 1] && GtQ[n, 1]
 

Maple [F]

\[\int {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{\frac {5}{2}}d x\]

Input:

int((a+b*arccos(d*x^2+1))^(5/2),x)
 

Output:

int((a+b*arccos(d*x^2+1))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arccos(d*x^2+1))^(5/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=\int \left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a+b*acos(d*x**2+1))**(5/2),x)
 

Output:

Integral((a + b*acos(d*x**2 + 1))**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(d*x^2+1))^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imagi 
nary; found sqrt((-_SAGE_VAR_d*_SAGE_VAR_x^2)-2)
 

Giac [F]

\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=\int { {\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+b*arccos(d*x^2+1))^(5/2),x, algorithm="giac")
 

Output:

integrate((b*arccos(d*x^2 + 1) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^{5/2} \,d x \] Input:

int((a + b*acos(d*x^2 + 1))^(5/2),x)
 

Output:

int((a + b*acos(d*x^2 + 1))^(5/2), x)
 

Reduce [F]

\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^{5/2} \, dx=\left (\int \sqrt {\mathit {acos} \left (d \,x^{2}+1\right ) b +a}d x \right ) a^{2}+2 \left (\int \sqrt {\mathit {acos} \left (d \,x^{2}+1\right ) b +a}\, \mathit {acos} \left (d \,x^{2}+1\right )d x \right ) a b +\left (\int \sqrt {\mathit {acos} \left (d \,x^{2}+1\right ) b +a}\, \mathit {acos} \left (d \,x^{2}+1\right )^{2}d x \right ) b^{2} \] Input:

int((a+b*acos(d*x^2+1))^(5/2),x)
 

Output:

int(sqrt(acos(d*x**2 + 1)*b + a),x)*a**2 + 2*int(sqrt(acos(d*x**2 + 1)*b + 
 a)*acos(d*x**2 + 1),x)*a*b + int(sqrt(acos(d*x**2 + 1)*b + a)*acos(d*x**2 
 + 1)**2,x)*b**2